Guide to vacuum physics and engineering

Many applications require vacuum conditions, including sensitive physics experiments, particle accelerators, electron microscopes, and highly-precise lasers. Vacuum conditions remove interference due to atmospheric fluctuations and allow for unimpeded travel at nearly-unlimited speeds. These applications all require the use of a vacuum pump to achieve near-vacuum conditions, and a vacuum chamber to contain the vacuum. We cover the basic theory and engineering behind both devices here.

Types of vacuum

We must first distinguish between the different types of vacuum. An ideal vacuum is an unachievable state, but it is possible to get very close to an ideal vacuum using the most advanced vacuum pumps and vacuum chambers. Depending on the quality of the vacuum (i.e. how close the vacuum approaches the theoretical ideal) we classify artificial vacuums into different types, as shown in the below table¹:

Type of vacuum	Pressure	Equipment required
Low/mid vacuum	$>10^{-3}\text{ torr}$	Mechanical pump (e.g. rotary-vane)
High vacuum	$10^{-3}-10^{-6}\text{ torr}$	Mechanical pump (e.g. rotary-vane)
Ultra-high vacuum	$10^{-12}-10^{-7}\text{ torr}$	Mechanical + turbomolecular pump
Extreme-high vacuum	$\le 10^{-12}\text{ torr}$	Mechanical + turbomolecular pump + ion pump

Note: Some sources quote $10^{-3}\mathrm{torr}$ as the cutoff for high vacuum whereas some other sources quote $10^{-4}\mathrm{torr}$; we have elected to go with the former.

Ideal pump theory

Vacuum pumps evacuate air out of a vacuum chamber at an exponential rate, with the exponential decrease in absolute pressure $P(t)$ described by the following equation:

$$P(t)=P_0{e}^{-rt/V}$$

Where $V$ is the volume of the vacuum chamber (assumed to be constant, for a cylinder it would be $V=2\pi Rh$ where $R$ is the radius and $h$ is the height) and $r$ is the pump’s flow rate (in units of volume over time, e.g. ${\mathrm{ft}}^3/\min$), and $P_0=\text{1 atm}$ is the initial pressure (by definition equal to standard atmospheric pressure). It is not hard to see that this corresponds to the differential equation $P^{\prime }=-\frac{r}{V}P$ for constant $r$ and $V$, with $P(0)=P_0$. Upon rearrangement, the depressurization time required to reach a pressure of $P$ is given by:

$$t=\frac{V}{r}\ln \frac{P_0}{P}=\tau \ln \left(\frac{P_0}{P}\right),\tau \equiv \frac{V}{r},P\le P_0$$

Where $\tau$ is a characteristic time constant for the vacuum system. Note that the $\propto \ln (1/P)$ dependence means that getting to progressively higher vacuums (equivalent to lower ultimate pressures) will take an increasing amount of time. At $t\approx 4.7\tau$ the vacuum pressure will be less than 1% of atmospheric pressure, while at $t\approx 9.2\tau$ the vacuum pressure will be less than 0.01% of atmospheric pressure. Assuming a vacuum pump is capable of achieving high vacuum ($\approx 10^{-3}\text{ torr}$), it will achieve this at $t\approx 13.5\tau$.

Non-ideal pump theory

We should note that the above two formulas are only really valid for a constant flow rate $r$. If the flow rate was a general function of time, we would need to solve the more general nonlinear differential equation:

$$\frac{dP}{dt}=-\frac{r(t,P)}{V}P(t)$$

For a flow rate $r(t,P)$ that is initially close to constant but decreases greatly for low absolute pressures, we propose the following ansatz:

$$r(P)=\frac{2r_0}{1+\exp (k_{p}P)},k_p\ll 1$$

Where $r_0$ is the constant flow rate given by the vacuum pump manufacturer (typically in units of $\mathrm{L}/\min$ or ${\mathrm{m}}^3/\mathrm{hr}$ or $\mathrm{f}{\mathrm{t}}^3/\min$), and $k_P$ is a dimensionless constant that is close to zero. For high absolute pressures (e.g. close to atmospheric pressure) $r\approx r_0$, while for low absolute pressures $r$ can be significantly smaller than $r_0$, leading to a lower pump rate. This is a sigmoid curve and frequently used as a first approximation for population dynamics in the case of limited resources. For small $k_p$ the decay is very slow, making it a suitable model for vacuum depressurization.

We may now substitute in this ansatz into the depressurization ODE. Performing a change of variables $\tilde{P}\equiv k_{p}P$ allows us to write the ODE in the form:

$$\frac{d\tilde{P}}{dt}=-\frac{\alpha \tilde{P}}{1+e^{\tilde{P}}},\alpha \equiv \frac{2r_0}{V}$$

This differential equation possesses no exact solution in terms of elementary functions. However, we may make a Taylor approximation $e^{\tilde{P}}\approx 1+\tilde{P}$ (which is valid since $k_p\ll 1$), giving us the following simplified ODE, which can be solved analytically:

$$\frac{d\tilde{P}}{dt}\approx -\frac{\alpha \tilde{P}}{2+\tilde{P}}$$

With the initial condition $\tilde{P}(0)=k_p{P}_0$ we obtain an implicit solution of:

$$\tilde{P}=2\ln \left(k_p{P}_0\right)+k_p{P}_0-\alpha t-2\ln \tilde{P}$$

Or in terms of the absolute pressure $P$:

$$P=\frac{2\ln \left(k_p{P}_0\right)-\alpha t}{k_p}+P_0-\frac{2\ln (k_{p}P)}{k_p},k_p>0$$

This solution can also be written in the following form:

$$P=A+Bt+C\ln (k_{p}P),k_p>0$$

Where $A=\frac{2\ln (k_p{P}_0)}{k_p}$, $B=-\frac{\alpha }{k_p}$, and $C=-2/k_p$. It diverges in the limit $k_p\to 0$, and since we made the assumption that $k_p\ll 1$ it does not hold for large values of $k_p$. For $t\to \infty$, $P$ decays to zero, just like the idealized solution, but the decay is far slower. Hence, while it may look like the idealized solution on the surface, its behavior diverges from the idealized solution after a short period of time.

Particle transport and in vacuum chambers

One of the most common uses of vacuum chambers is in particle accelerators, whether linear accelerators (linacs), synchrotrons, gyrotrons, among other machines. As these machines need to accelerate charged particles to relativistic speeds, they cannot have any air within, which would lead to energy losses and loss of collimation, as particles collide with air molecules, causing them to scatter in all directions.

To quantify the quality of a vacuum chamber in microscopic terms, a common metric is the mean free path. The mean free path is a rough measure of how far the electrons will be able to travel through a medium without scattering. For a gaseous medium (specifically an ideal gas), the mean free path $\ell$ is given by:

$$\ell =\frac{k_{B}T}{\sqrt{2}\pi d^{2}p}$$

Where $k_B$ is the Boltzmann constant, $T$ is the temperature, $d$ is the kinetic diameter of a particle, and $p$ is the pressure. The kinetic diameter of various particles is listed in the table below (based on this Wikipedia article and NOAA), according to their respective concentrations in air:

Molecule	Kinetic diameter	Mole fraction in air
Nitrogen	364 pm	78.084%
Oxygen	346 pm	20.946%
Water (vapor)	265 pm	0% (ideal), <4% (typical)
Argon	340 pm	0.934%
Carbon dioxide	330 pm	0.042%
Neon	275 pm	0.0018%
Ozone	460 pm	0.0005%
Methane	380 pm	0.0002%
Krypton	360 pm	0.0001%
Hydrogen	289 pm	<0.0001%

Note: The following is only true for perfectly dry air. Typically, water vapor occupies a significantly higher fraction of air, which can be up to 4% (according to NOAA), although in desert climates air can become nearly perfectly dry.

Meanwhile, the IMFP formula gives the mean free path for a solid (assuming the particles are electrons) as follows²:

$$\frac{1}{\ell }=\frac{1}{\pi E}{\int }_{{\omega }_{min}}^{{\omega }_{max}}d\omega {\int }_{k_{-}}^{k^{+}}\frac{dk}{k}\mathrm{Im}\left(-\frac{1}{ϵ(k,\omega )}\right),k_{\pm }=\sqrt{2E}\pm \sqrt{2(E-\omega )}$$

Where $E$ is the (kinetic) energy of the incident electrons, $ϵ(k,\omega )$ is the dielectric function of the solid, $\omega$ is the (angular) frequency (in natural units this is equal to the energy transfer since $E=\hslash \omega \to \omega$) where ${\omega }_{min}$ is the minimum energy lost, ${\omega }_{max}$ is the maximum energy lost, and one integrates between the smallest momentum transfer $k_{-}$ and the largest $k^{+}$ (in natural units, $p=k$ and $E=m_e$ since $\hslash =c=1$). We may use a series of successive approximations based on the electron energy:

$$\frac{1}{\ell }\approx \{\begin{array}{ll}\frac{(4+0.44Z^{0.5}+0.104E^{0.872})a^{1.7}}{Z^{0.3}(1-W)},& 0.1\mathrm{keV}\le E\le 50\mathrm{keV}\\ \frac{\alpha E}{E_p^2\left(\beta \ln (\gamma \alpha E)-\frac{C}{E}+{\left(\frac{D}{E}\right)}^2\right)},& 50\mathrm{keV}\le E\le 200\mathrm{keV}\end{array}$$

Where:

• $m_e$ is the electron mass
• $a^3=\frac{10^{21}M}{\rho N_A(g+h)}$ is the average atomic spacing, in units of $\mathrm{n}{\mathrm{m}}^3$
• $M$ is the atomic (or molecular) weight
• $\rho$ is the mass density (in units of $\mathrm{g}/\mathrm{c}{\mathrm{m}}^3$)
• $N_A$ is Avogadro’s number
• $Z$ is the atomic number (or average atomic number for a compound)
• $W=0.06H=0.02E_g$, where $E_g$ is the bandgap energy (in $\mathrm{eV}$) and (if a compound) $H$ is the heat of formation of the compound, in units of eV per atom
• $\alpha (E)=\frac{1+r/2}{(1+r{)}^2}$ where $r\equiv E/(m_e{c}^2)=E/0.511\mathrm{MeV}$ is the ratio between the electron energy and the electron mass energy ($0.511\mathrm{MeV}$)
• $E_p=28.816\sqrt{U}$ and has units of $\mathrm{eV}$, where $U=\frac{N_{\nu }\rho }{M}$ and $N_{\nu }$ is the number of valence electrons per atom (or molecule, for compounds)
• $\gamma =0.191{\rho }^{-1/2}$ and has units of $\mathrm{e}{\mathrm{V}}^{-1}$
• $\beta =-1+\frac{9.44}{(E_p^2+E_g^2{)}^{1/2}}+0.69{\rho }^{0.1}$ and has units of $\mathrm{e}{\mathrm{V}}^{-1}\mathrm{n}{\mathrm{m}}^{-1}$
• $C=19.7-9.1U$ and has units of $\mathrm{n}{\mathrm{m}}^{-1}$
• $D=534-208U$ and has units of $\mathrm{e}{\mathrm{V}}^{-1}\mathrm{n}{\mathrm{m}}^{-1}$

Note: Of course, the electrons can still travel after scattering, but the loss of energy and change in momentum due to scattering events will cause particle beams to diverge and lose collimation, which is highly undesirable.

We will not cover the full details of particle transport and collisions here as it is a highly-complex topic. A more in-depth overview can be found in Theoretical analysis of electron beam scattering.

Footnotes

1. Most of the data for the table comes from https://en.wikipedia.org/wiki/Orders_of_magnitude_(pressure) as well as https://www.leybold.com/en-us/knowledge/vacuum-fundamentals/vacuum-generation/ultra-and-extreme-high-vacuum. ↩

2. From the Wikipedia article on the subject ↩